Research Problems in Function Theory — Problem 8.6

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For $p>0$, the Bergman space $A^p$ consists of holomorphic $f$ on $\mathbb D$ with $\int_{\mathbb D}|f|^p\,dA<\infty$. A zero divisor is the multiset of zeros of a nonzero such function, with multiplicities and no accumulation inside the disc. Any finite multiplicity at $0$ is separated out before taking reciprocal products; write the remaining zeros $z_k\ne0$ in nondecreasing modulus. For a finite nonempty $F\subset\mathbb T$ let $I_\nu$ be the complementary open arcs and $|I_\nu|$ their angular lengths. Define \[R(I)=\{re^{i\theta}:1-|I|/(2\pi)\le r<1,\ e^{i\theta}\in\overline I\}, \quad G_F=\mathbb D\setminus\bigcup_\nu R(I_\nu), \] \[\kappa(F)=\frac1{2\pi}\sum_\nu\frac{|I_\nu|}{2\pi} \log\big(2\pi/|I_\nu|+1\big). \] Sums over zeros in $G_F$ count multiplicity.

Statement recorded in the supplied source.

Characterize zero divisors for each fixed $A^p$, or at least give a nontrivial converse to the necessary estimate \[\prod_{k=1}^N|z_k|^{-1}=O(N^{1/p+\varepsilon})\quad(N\to\infty) \quad\text{for every }\varepsilon>0, \] whose exponent cannot generally omit $\varepsilon$. The source update supplies a modified Korenblum criterion: for an absolute $\lambda>1$, finiteness of \[\sup_F\big\{\sum_{z_k\in G_F}(1-|z_k|)-\lambda p^{-1}\kappa(F)\big\} \] is necessary, while finiteness with $\lambda$ replaced by $\lambda^{-1}$ is sufficient. It also states that \[\sup_F\kappa(F)^{-1}\sum_{z_k\in G_F}(1-|z_k|)<\infty \] characterizes being a zero divisor for some $A^p$, $p>0$. These two nearby fixed-$p$ bounds do not replace the request for an exact fixed-$p$ characterization. [S2]

Short English statement

Characterize Bergman-space zero sequences at each exponent, or find meaningful converse conditions to the reciprocal-product bound, preserving the source’s necessary and sufficient comparison criteria.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2308006 (AMR-022-8006). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Walter K. Hayman and Edward F. Lingham, Research Problems in Function Theory (2018), Problem 8.6, its complete update and relevant preceding definitions. https://arxiv.org/abs/1809.07200.